Spin fluctuations in nearly magnetic metals from ab initio dynamical spin susceptibility calculations: Application to Pd and Cr_ {95} V_ {5}

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Spin fluctuations in nearly magnetic metals from ab-initio

dynamical spin susceptibility calculations: application to Pd and

Cr95V5

J.B.Staunton†, J.Poulter‡, B.Ginatempo¶, E.Bruno¶and D.D.Johnson§

† Department of Physics, University of Warwick, Coventry CV4 7AL, U.K.

‡ Department of Mathematics,Faculty of Science, Mahidol University,Bangkok 10400, Thailand.

¶ Dipartimento di Fisica and Unita INFM, Universita di Messina, Italy

§ Department of Materials Science and Engineering, University of Illinois, IL 61801, U.S.A.

(February 1, 2008)

Abstract

We describe our theoretical formalism and computational scheme for mak-

ing ab-initio calculations of the dynamic paramagnetic spin susceptibilities of

metals and alloys at finite temperatures. Its basis is Time-Dependent Den-

sity Functional Theory within an electronic multiple scattering, imaginary

time Green function formalism. Results receive a natural interpretation in

terms of overdamped oscillator systems making them suitable for incorpora-

tion into spin fluctuation theories. For illustration we apply our method to the

nearly ferromagnetic metal Pd and the nearly antiferromagnetic chromium al-

loy Cr95V5. We compare and contrast the spin dynamics of these two metals

and in each case identify those fluctuations with relaxation times much longer

than typical electronic ‘hopping times’.

75.40Gb,75.10Lp,71.15Mb,75.20En

Typeset using REVTEX

1

There is renewed interest in spin fluctuations in materials close to magnetic order. This

is due in part to a realisation that nearly critical magnetic fluctuations may be important

factors governing the non-conventional properties of a wide range of materials which in-

clude the high Tc superconducting cuprates and heavy fermion systems [1]. The strongly

correlated electrons in many of these systems, however, have meant that most theoretical

work has concentrated on parameterised models in which the electronic motion is treated

rather simply. Another complementary approach is to use an ab initio theory such as Time

Dependent Density Functional Theory (TDDFT) [2] but apply it to materials where it can

be expected to work i.e. where the effects of electron correlations are not so important,

but which otherwise have important similarities to the systems in question. For example,

with its perovskite structure containing transition metal(TM)-oxygen planes, Sr2RuO4 has

several aspects in common with the HTC materials. But the presence of the 4d TM Ru

rather than the narrower band 3d TM Cu means that electron correlation effects are smaller

and therefore DFT-based calculations can provide a valuable starting point. Moreover, its

exotic superconductivity at low temperature seems likely to be affected by spin fluctuations

[3–5]. Concerning another example, the transition temperature separating paramagnetic

and magnetically ordered phases of the cubic transition metal compound MnSi, which has

the B20 crystal structure, is driven down to zero temperature upon the application of pres-

sure. In the vicinity of the critical pressure for this quantum phase transition the system

exhibits non- or marginal Fermi liquid properties [6]. In this paper we investigate the energy

and temperature dependence of spin fluctuations in two systems which although structurally

and compositionally simpler than the two mentioned above, are also close to magnetic phase

transitions. The first is palladium which is nearly ferromagnetic and the second is the nearly

anti-ferromagnetic Cr95V5 alloy.

Theoretical models in which an effective action for the slow spin fluctuations is written

down have contributed greatly to our understanding of the properties of itinerant electron

systems close to magnetic order [6,7]. Recent work which can incorporate results from

DFT-based ‘Fixed Spin Moment’ (FSM) electronic structure calculations, treats these fluc-

2

tuations classically [8]. A Landau-Ginzburg-like energy functional is written down, a free

energy constructed which includes terms describing the interactions between the fluctuations

(the mode-mode coupling) and properties such as the static susceptibility, specific heat and

resistivity calculated. The FSM electronic structure calculations can be used to determine

the coefficients in this functional [8]. Many informative studies have been carried out. For

those calculations with such a DFT basis, these still remain qualitative investigations be-

cause of the lack of a prescription for the effective number of modes to include in the theory

and its variation with temperature [6,7]. Stoner single-particle excitation effects are also

largely ignored. Both these issues can be addressed by the development and application of

methods to calculate the temperature-dependent, dynamic paramagnetic spin susceptibility

of nearly magnetic materials. The development of dynamic susceptibility calculations is

particularly pertinent now that inelastic neutron scattering experiments, such as the time of

flight measurements, have developed to the extent that spin fluctuations in nearly magnetic

metals can be accurately measured [9].

We have recently devised and proven a new scheme for calculating the wave-vector and

frequency dependent dynamic spin susceptibility of metallic systems [10] which is based on

the Time Dependent Density Functional Theory (TDDFT) of Gross et al. [2] and as such is

an all electron theory. This enables the temperature dependent dynamic spin susceptibility

of metals and compositionally disordered alloys to be calculated.

Over the past few years great progress has been made in establishing TDDFT [2]. Analogs

of the Hohenberg-Kohn [11] theorems of the static density functional (DFT) formalism have

been proved and rigorous properties found. By considering a paramagnetic metal subjected

to a small, time-dependent external magnetic field, b(r, t) which induces a magnetisation

m(r, t) and using TDDFT in [10] an expression for the dynamic paramagnetic spin sus-

ceptibility χ(q, w) via a variational linear response approach can be derived [12]. Accurate

calculations of dynamic susceptibilities from this basis have been scarce (e.g. [13]) because

they are difficult and computationally demanding. In ref. [10] we showed that these prob-

lems can be mitigated by accessing χ(q, w) via the corresponding temperature susceptibility

3

χ(q, wn) where wn denotes a bosonic Matsubara frequency [14]. We demonstrated this ap-

proach by an investigation into the nature of the spin fluctuations in paramagnetic Cr and

compositionally disordered Cr95V5 and Cr95Re5 alloys with good agreement with experi-

mental data [10]. For example, recent inelastic neutron scattering experiments [9,15] have

measured incommensurate AF ‘paramagnons’, persisting up to high frequencies in the latter

system which were also shown in our calculations

In [10] we provided a brief outline of this approach for crystalline systems with one atom

per unit cell only so in the next section we provide its generalisation to multi-atom per unit

cell materials in which one or more sublattices may be substitutionally disordered. We also

describe the techniques involved in the method in more detail. In the following section we

discuss our study of the frequency, wave-vector and temperature dependence of the spin

fluctuations in Pd and identify the paramagnon regime. We then move on to carry out a

similar study of the incommensurate antiferromagnetic spin fluctuations in the dilute Cr95V5

alloy. In each case the results can be interpreted simply in terms of an over-damped harmonic

oscillator model [6]. In the latter case we show how the parameters scale with temperature

and compare and contrast the two systems. In the final section we draw some conclusions

and remark how this work has the potential to be integrated into spin fluctuational theories

of nearly and weakly magnetic metals [6].

I. THE DYNAMICAL SPIN SUSCEPTIBILITY χ(Q,W ).

The equilibrium state of a paramagnetic metal, described by standard DFT, has density

ρ0(r) and its magnetic response function

χ(rt; r′t′) =δm[b](r, t)

δb(r′, t′)|b=0,ρ0

(1.1)

is given by the following Dyson-type equation.

χ(rt; r′t′) = χs(rt; r′t′) +

∫dr1

∫dt1

∫dr2

∫dt2χs(rt; r1t1)Kxc(r1t1; r2t2)χ(r2t2, r

′t′) (1.2)

4

χs is the magnetic response function of the Kohn-Sham non-interacting system with the

same unperturbed density ρ0 as the full interacting electron system, and

Kxc(rt; r′t′) =

δbxc(r, t)

δm(r′, t′)|b=0,ρ0

(1.3)

is the functional derivative of the effective exchange-correlation magnetic field with respect

to the induced magnetisation. As emphasised in ref. [2] eq.1.2 represents an exact represen-

tation of the linear magnetic response. The corresponding development for systems at finite

temperature in thermal equilibrium has also been described [12]. In practice approximations

to Kxc must be made and this work employs the adiabatic local approximation (ALDA) [2]

so that

KALDAxc (rt; r′t′) =

dbLDAxc (ρ(r, t), m(r, t))

dm(r, t)|ρ0,m=0δ(r − r′)δ(t − t′)

= Ixc(r)δ(r− r′)δ(t − t′) (1.4)

On taking the Fourier transform of eq.1.2 with respect to time we obtain the dynamic spin

susceptibility χ(r, r′; w).

For computational expediency we consider the corresponding temperature susceptibil-

ity [14] χ(r, r′; wn) which occurs in the Fourier representation of the temperature function

χ(rτ ; r′τ ′) that depends on imaginary time variables τ ,τ ′ and wn are the bosonic Matsubara

frequencies wn = 2nπkBT . Now χ(r, r′; wn) ≡ χ(r, r′; iwn) and an analytical continuation

to the upper side of the real w axis produces the dynamic susceptibility χ(r, r′; w).

We define our general system as having a crystal structure with lattice vectors Ri and

where there are Ns non-equivalent atoms per unit cell. On the lth of the Ns sublattices there

are Nl atomic species with concentrations c(l)αl

(αl = 1, · · · , Nl). In each unit cell the Ns

atoms are situated at locations al, l = 1, · · · , Ns. The volume of the unit cell is VWS. On

carrying out a lattice Fourier transform over the lattice vectors Ri we obtain the following

Dyson equation for the temperature susceptibility

χαl(xl,x′

l′,q, wn) =N

l′∑γ

l′

cl′

γl′χαlγl′

s (xl,x′

l′,q, wn)

5

+Ns∑l′′

∫dx′′

l′′

Nl′′∑

γl′′

cl′′

γl′′

χαlγl′′

s (xl,x′′

l′′ ,q, wn)Iγ

l′′

xc (x′′

l′′)χγ

l′′ (x′′

l′′ ,x′

l′,q, wn) (1.5)

with xl,x′

l′ and x′′

l′′ measured relative to atomic cells centred on al, al′ and al′′ respectively.

χαlγl′

s describes the non-interacting susceptibility averaged over all configurations in which

one site on sublattice l is occupied by an αl atom and another site on sublattice l′ is occupied

by a γl′ atom. χαl is the full susceptibility averaged over all configurations where an αl atom

is located on one site on sublattice l.

In terms of the lattice Fourier transform of the DFT Kohn-Sham Green function of the

static unperturbed system the non-interacting susceptibility can be written approximately

[18]

χαlγl′

s (xl,x′

l′ ,q, wn) =

−1

β

∑m

∫dk

vBZ

〈G(xl,x′

l′,k, µ + iνm)〉αlγl′〈G(x′

l′,xl,k + q, µ + i(νm + wn))〉γl′

αl(1.6)

where the integral is over the Brillouin zone of the lattice and k, q and k + q are all

wavevectors within this Brillouin zone which has volume vBZ . µ is the chemical potential

and νm is a fermionic Matsubara frequency (2n + 1)πkBT .

The Green function can be obtained within the framework of multiple scattering (KKR)

theory [16] and this makes this formalism applicable to disordered alloys as well as ordered

compounds and elemental metals, the disorder being treated by the Coherent Potential

Approximation (CPA) [17]. Then the partially averaged Green function with species αl on a

site at Ri +al on the lth sublattice and species γl′ on a site at Rj +al′ on sublattice number

l′

〈G(Ri + al + xl,Rj + al′ + xl′, z)〉αl,γl′(1.7)

can be evaluated in terms of deviations from the Green function of an electron propagating

through a lattice of identical potentials determined by the CPA ansatz [18]. The lattice

Fourier transform used in equation (1.6) is expressed as

〈G(xl,x′

l′,k, z)〉αlγl′=

∑L,L′

Zαl

L (xl, z)〈τ l,l′

L,L′(k, z)〉αlγl′Z

γl′

L′ (x′

l′ , z)

6

+ δl,l′

∑L,L′

[δαlγl(Zαl

L (xl, z)〈τ l,lL,L′(z)〉αl

Zαl

L′ (x′

l, z) − Zαl

L (x<l , z)Jαl

L (x>l , z))

− Zαl

L (xl, z)〈τ l,lL,L′(z)〉αlγl

Zγl

L′(x′

l, z)] (1.8)

where Zαl

L and Jαl

L are respectively the regular and irregular solutions of the Schrodinger

equation in an atomic cell on sublattice l inhabited by an αl atom and L,L′ represent

the angular momentum quantum numbers. The lattice Fourier transform of the averaged

scattering path operator τ l,l′

L,L′(Ri − Rj, z) is specified by the following expressions [16]

〈τ l,l′

L,L′(k, z)〉αlγl′=

∑L1L2

Dlαl,LL1

(z)τ l,l′

L1,L2(k, z)Dl′

γl′

,L2L′ (1.9)

where, in matrix notation with respect to angular momentum L,(D describes the transpose)

Dlαl,LL1

(z) = [1 + τ ll(z)(mαl(z) − mc,l(z))]−1LL1

(1.10)

with mαl being the inverse of the scattering t-matrix tαl of an αl atom potential, mc,l the

inverse of the CPA t-matrix for the lth sublattice, tc,l and τ ll′ the unit cell-diagonal part of

the CPA scattering path operator. The following averages also involve these quantities

〈τ l,l′

L,L′(z)〉αlγl′=

∑L1L2

Dlαl,LL1

(z)τ l,l′

L1,L2(z)Dl′

γl′

,L2L′(z)

〈τ l,lL,L′(z)〉αl

=∑L1

Dlαl,LL1

(z)τ l,lL1,L′(z) (1.11)

Finally the lattice Fourier transform of this CPA scattering path operator is given by

τ ll′

L,L′(k, z) = tc,lL,L′(z)δll′ +∑l′′

∑L1L2

tc,lL,L1(z)gll′′

L1L2(k, z)τ l′′l′

L2L′(k, z) (1.12)

and gll′

L1L2(k, z) are the structure constants for the crystal structure [19].

To solve equation (1.5), we use a direct method of matrix inversion in which, for example,

χs is cast into matrix form of order (∑Ns

l=1 SlNl)×(∑Ns

l=1 SlNl) where Sl is the number of spatial

grid points associated with the lth sublattice. Local field effects are thus fully incorporated.

The full Fourier transform

χ(q,q; wn) = (1/VWS)∑

l

∑αl

clαl

∑l′

eiq·(al−al′)∫

dxl

∫dx′

l′eiq·(xl−x

′

l′)χαl(xl,x

′

l′,q, wn) (1.13)

7

can then be constructed.

The most computationally demanding parts of the calculation are the convolution inte-

grals over the Brillouin Zone which result from the expression for χs, eq. (1.6) i.e.

∫dkτ ll′

LL1(k, z)τ l′l

L2L3(k + q, z′) (1.14)

Since all electronic structure quantities are evaluated at complex energies z, these con-

volution integrals have no sharp structure and can be evaluated straightforwardly by an

application of the adaptive grid method of B.Ginatempo and E.Bruno [20] which has been

found to be very efficient and accurate. In this method one can preset the level of accuracy

of the integration by supplying an error parameter ǫ. In the calculations described in this

paper we have used ǫ = 10−4 so that we obtain 4 significant figure accuracy on χs. This

control on accuracy is crucial for the description of the long wavelength paramagnons in Pd.

In this case we needed to sample around 2000 k-points at energies on the contour distant

0.5Ry. off the real axis but up to 50000 k-points very close (0.001 Ry.) to the real energy

axis. Around half that number of points were required for the Cr95V5 calculations.

We evaluate the Matsubara sum shown in equation (1.6) by using suitable complex

energy contours to enclose some of the Matsubara poles and exploiting the hermiticity of

the single-electron Green function G(r, r′, z). Consequently a quantity like χs at a positive

bosonic Matsubara frequency wn with form

A(r, r′, wn) = −1

β

∑m

G(r, r′, µ + iνm)G(r′, r, µ + iνm + iwn) (1.15)

where the fermionic Matsubara sum is over m = 0,±1,±2, · · · ,±∞ can be rewritten as

A(r, r′, wn) = −2

β

Mmax∑m=0

Re(G(r, r′, µ + iνm)G(r′, r, µ + iνm + iwn)) + (1.16)

1

πIm

∫C

dz (G(r, r′, z)G(r′, r, z + iwn))f(z) −1

β

n−1∑m=0

G(r, r′, µ − iνm)G(r′, r, µ + iwn − iνm)

Both sums are over positive fermionic frequencies only and f(z) is the Fermi function. C is

a contour which encloses the first Mmax fermionic Matsubara poles with the real axis. In our

calculations we have used a simple box contour whose top lies exactly halfway between two

8

neighbouring νm’s which makes the Fermi factor at this z real. The terms in the Matsubara

sums for νm’s > 0.001 are evaluated by interpolating from a grid ν = 0.001 × 100.1p, with

p = 0, 1, · · · , 30 or are evaluated directly. Mmax is chosen so νMmax≈ 1.

Once the temperature susceptibility χ(q,q; wn) has been calculated the dynamic sus-

ceptibility can be found. As discussed in ref. [14], for example, we can define the retarded

response function χ(q,q, z) of a complex variable z. Since it can be shown [14] formally

that limz→∞ χ(z) ∼ 1/z2 and we can obtain χ(iwn) from the above analysis it is possible

to continue analytically to values of z just above the real axis, i.e. z = w + iη. In order to

achieve this we fit our data to a function describing a set of overdamped oscillators, i.e.,

χ(q,q, wn) =∑M

χM(q)

(1 + (wn/ΓM(q)) + (wn/ΩM (q))2)(1.17)

in which M is an integer. From this the dynamic susceptibility can be written down and

the imaginary part becomes

Imχ(q,q, w) =∑M

χM (q)(w/ΓM(q))

[1 − (w/ΩM(q))2]2 + (w/ΓM(q))2. (1.18)

This form ensures that the sum rule involving the static susceptibility χ(q) is satisfied, i.e.

χ(q) =2

π

∫∞

0dw

Imχ(q,q, w)

w(1.19)

In nearly magnetic metals the nature of the spin fluctuations can be succinctly described

via the variance < m2(q) >. From the fluctuation dissipation theorem

< m2(q) >=1

π

∫∞

−∞

dwImχ(q,q, w)

(1 − e−βw)(1.20)

By introducing an energy cutoff in this expression the slower fluctuations can be revealed.

We find that very good fits are obtained with small M in equation (1.17) for a wide

range of wn’s ( < 500 meV) for the systems we have examined to date. For Cr and its dilute

alloys using M = 1 in equation (1.17) provides an excellent fit. This means that a close

analogy with an overdamped harmonic oscillator can be made. In the case of palladium,

equation (1.17) with M = 2 gives an excellent representation of χ(q,q, wn). Now the

spin fluctuations can be described in terms of two overdamped oscillators with two distinct

timescales. The slower one describes the famous paramagnons of palladium.

9

II. THE SPIN FLUCTUATIONS OF NEARLY FERROMAGNETIC PALLADIUM.

Although possessing one of the highest Fermi energy densities of states of all the tran-

sition metals, palladium has neither a magnetically ordered nor a superconducting ground

state. With its nearly filled d-bands, it is, however, on the verge of being a ferromagnet.

This is shown by its uniform paramagnetic spin susceptibility which is greatly enhanced

over that of an equivalent non-interacting system by exchange-correlation effects. Moreover,

the temperature dependence of the susceptibility also demonstrates effects of long-lived fer-

romagnetic spin-fluctuations or paramagnons. It has even been proposed that these spin

fluctuations might induce p-wave superconductivity in this material on account of similari-

ties with spin-fluctuation induced superfluidity in He3 [21].

Several theoretical studies of the spin fluctuations of Pd have been carried out. The early

ones were based on a jellium model for the electrons and examined the contribution that

spin fluctuations make to the specific heat and their influence on any superconducting phase

transition as well as describing the cross-section that would be measured in an inelastic

neutron scattering experiment [22]. Later work has attempted to remove the deficiencies

of such a single-band model and to describe the band structure effects more realistically

[13]. From his calculations of the inelastic neutron scattering cross-section for a range of

scattering angles and incident energies, Doniach [22] estimated the circumstances under

which the paramagnons of Pd might be observed. Very recently time of flight measurements

on a neutron spallation source have succeeded in detecting these long-wavelength, long-lived

spin fluctuations [9].

Using the formalism described in the last section, we have calculated the temperature

susceptibility of Pd at T = 100K and have then determined the dynamic susceptibility

χ(q, w). Currently our calculations use descriptions of the equilibrium states in terms of

scalar-relativistic atomic sphere approximation (ASA) effective one-electron scattering po-

tentials and charge densities. The formalism, however, is applicable to full potentials and

non-spherically symmetric charge densities. The lattice spacing of 3.83 A which we used in

10

the Pd calculations was determined from the minimum of the ASA total energy and found

to be 1.5% smaller than the experimentally measured value. At 15.7 µ2BeV −1 our calculated

uniform susceptibility χ(0, 0) is somewhat lower than the value 24.3 µ2BeV −1 measured ex-

perimentally [9]. For small wavevectors q the inverse of our calculated static susceptibility

depends on the magnitude squared of q, i.e. χ−1(q, 0) ≈ χ−1(0, 0)+ cq2 as also found exper-

imentally and in other calculations [13]. We obtain a value of approximately 900 µ2BA2meV

for c whereas the value extracted from inelastic neutron scattering experiments is 294 ± 130

µ2BA2meV . Since palladium is so close to a ferromagnetic phase transition both χ(0, 0) and

c are very sensitive to the value of the lattice parameter used and the RPA approximation

used for the exchange-correlation interactions. Using the experimental lattice spacing in the

calculations greatly increases χ(0, 0) (to 23.5 µ2BeV −1 close to the value determined from

experiment) and both χ(0, 0) and c are likely to be altered if spin fluctuation interaction

effects are treated beyond the LDA [7]. Key aspects of the dynamic behaviour turn out not

to be so sensitive.

In figure 1 we show the calculated imaginary part of χ(q, w) for a range of frequencies up

to 200 meV for selected wavevectors q along the (1, 0, 0) direction. The sharp peaks at low

frequencies for q < 0.1 indicate paramagnon-like behaviour. This is illustrated further in

figure 2 which shows the magnetic correlations < m2(q) > using energy cutoffs of 500meV

and 50meV. Once again the paramagnons are clearly visible for a narrow region of small

wave-vectors q.

We find that the results can be interpreted in terms of two overdamped simple harmonic

oscillators each with a characteristic timescale. The longer timescale one encapsulates the

paramagnon behaviour. Imχ(q,q, w) can now be written in the following way

Imχ(q,q, w) ≈χ1(q)(w/Γ1(q))

(1 + (w/Γ1(q))2)+

χ2(q)(w/Γ2(q))

(1 + (w/Γ2(q))2)(2.1)

Figure 3(a) shows the quantities χ1(q), χ2(q) and their sum whereas figure 3(b) shows

Γ−11 (q) and Γ−1

2 (q). Evidently at small q the slow relaxation process is dominant and

h/Γ1(q) describes the long time taken for a spin density perturbation to relax back to the

11

equilibrium value. Moreover, we find that the product of χ1(q) with Γ1(q) varies as γ0q for

small q, so that spin density disturbance decays over a time proportional to the wavelength,

i.e. exhibits Landau damping [6]. We find γ0 to be 2.13 µ2BA which is in good agreement

with the experimentally determined value of 1.74 ± 0.8 [9] and evidently fairly insensitive

to the treatment of exchange-correlation effects.

III. INCOMMENSURATE ANTIFERROMAGNETIC SPIN FLUCTUATIONS IN

CR95V5.

Chromium, an early 3d transition metal, loses its anti-ferromagnetic ground state when

electrons are removed as a suitable dopant is added. For example a strongly exchange-

enhanced paramagnet exhibiting anti-ferromagnetic paramagnons is formed when just a few

atomic percent of chromium are substituted by vanadium. In this section we explore the

similarities and differences between these excitations in one such system, Cr95V5, and those

reported for palladium. We also examine the temperature dependence.

There is an extensive literature on chromium as the archetypal itinerant anti-ferromagnet

(AF) and its alloys [15]. It is well known that Cr’s famous incommensurate spin density

wave (SDW) ground state is linked to the nesting wave-vectors qnest identified in the Fermi

surface. This feature also carries over to its dilute alloys which have a range of AF proper-

ties [15] (see figure 4). Indeed the paramagnetic states of some of these alloys have recently

acquired a new relevance on account of analogies drawn with the high temperature su-

perconducting cuprates especially (LacSr1−c)2CuO4 [23]. Starting with ‘parent’ materials

Cr95Mn5 or Cr95Re5 which are simple commensurate AF materials and then lowering the

electron concentration by suitable doping causes the Neel temperature, TN , to drop rapidly

to zero. The paramagnetic metal which forms for dopant concentrations slightly in excess

of the critical concentration for this quantum phase transition is characterised by incom-

mensurate paramagnetic spin fluctuations. In [10] we described the spin fluctuations in the

paramagnetic state of three representative systems Cr95Re5, Cr and Cr95V5. We tracked

12

the tendency for the spin fluctuations to become more nearly commensurate with increas-

ing frequency and also with increasing temperature [10] as observed in experimental data.

Our estimates of TN in Cr95Re5 and Cr of 410K and 280K, respectively, were also in fair

agreement with the experimental values of 570K and 311K.

Although there have been several simple parameterised models to describe the magnetic

properties of Cr and its alloys [24], these have all concentrated on the approximately nested

electron ‘jack’ and slightly larger octahedral hole pieces of the Fermi surface (FS) [15]. These

sheets can be seen clearly in figure 4 which shows the Fermi surface of Cr95V5. The FS of

Cr95Re5’s is close to being perfectly nested and we calculated in [10] its spin fluctuations

above TN to be nearly commensurate for small frequencies w. We also looked at Cr and

Cr95V5. Cr’s and Cr95V5’s Fermi surfaces are progressively worse nested than Cr95Re5’s.

In our calculations for these two systems [10] we found the dominant slow spin fluctuations

to be incommensurate with wave-vectors equal to the FS nesting vectors qnest. Cr95V5’s is

shown on figure 4. At best, simple parameterised models only include the effects of all the

remaining electrons via an electron reservoir. Whilst finding the obvious similarities from

the FS basis between our results and results from such models, we showed that a complete

picture is obtained only when an electronic band-filling effect which favours a simple AF

ordering at low temperature is also considered.

As for the case of Pd we found that the spin fluctuations in the paramagnetic state of the

three systems are given an accurate description in terms of a overdamped diffusive simple

harmonic oscillator model. Here, however, a single channel is sufficient i.e. the susceptibility

closely fits the following

Imχ(q,q, w) ≈χ(q)(w/Γ(q))

(1 + (w/Γ(q))2)(3.1)

In this section we concentrate on the exchange-enhanced paramagnet Cr95V5 and focus

in particular on the temperature dependence of the incommensurate spin fluctuations and

show how the parameters of the oscillator model scale with temperature.

Figure 1(a) of reference [10] showed Imχ(q,q, w) for wavevectors q along the 0, 0, 1

13

direction for frequencies up to 500meV. It showed, in agreement with experiment [9], in-

commensurate AF paramagnons persisting up to high frequencies with intensity comparable

to that at low w. This is in striking contrast to the paramagnons of Pd. Figure 5 shows

this persistence of the spin fluctuations in Cr95V5 to high frequencies in a concise way. It

shows the magnetic correlations < m2(q) > at 100K for the same two energy cutoffs 50

and 500meV as was used to obtain figure 2 for palladium. Unlike those in the nearly fer-

romagnetic metal, the magnetic correlations grow by more than an order of magnitude for

all the wave-vectors shown as the energy cutoff is extended by a similar factor. The varia-

tion with temperature of the dynamic susceptibility and the magnetic correlations have also

been investigated. Figure 6 shows < m2(q) > calculated with a energy cutoff of 500meV

for 50K, 300K and 600K. There is a general trend for the weight to transfer towards the

commensurate wavevector (0, 0, 1) with increasing temperature. For a smaller energy cutoff

< m2(q) >, integrated over all q, decreases with increasing temperature.

Figures 7(a) and (b) show that both the static susceptibility χ(q) and the spin relaxation

time Γ−1(q)/h also peak at the incommensurate nesting vector (0, 0, 0.9) at low tempera-

ture and that weight is shifted towards (0, 0, 1) at higher temperatures. Figure 7(b) when

compared with the analogous figure 3(b) for Pd reveals the main difference between the

dominant spin fluctuations in the nearly antiferromagnetic Cr95V5 and the nearly ferromag-

netic Pd. For the dominant spin fluctuations, the spin relaxation times are some 50 times

faster. The small Q behavior is different also. (Q = q − q0, where q0 = 0 for Pd and an

incommensurate wave-vector for Cr95V5.) The product γ(q) of χ(q) and Γ(q) tends to a

constant for values of q close to q0 for Cr95V5 (consistent with ideas that the dynamical

critical exponent [6] should be 2 for antiferromagnetic itinerant electron systems) unlike the

linear variation of γ(q) with |q| for small q found in Pd. Between 50K and 300K, γ(q)

shows a weak variation with temperature where it has values of roughly 4 µ2B and 6.5 µ2

B at

the incommensurate wave-vector q0 and q = (0, 0, 1) respectively. Above 300K, γ(q) loses

the trough around q0 and by 600K it is roughly constant at 5.5 µ2B for the range of q from

q0 to (0, 0, 1).

14

IV. CONCLUSIONS

We have described the framework and method of implementation of our scheme for car-

rying out ab-initio calculations of the dynamic paramagnetic spin susceptibilities of solids at

finite temperatures. The approach is based upon Time-Dependent Spin Density Functional

Theory and is applicable to compositionally disordered alloys with multi-atom per unit cell

crystal structures. (We note here that this approach may also be adapted to the study of

magnetic excitations in magnetically ordered materials.) From an imaginary time multiple

scattering Green function formalism an expression for the temperature susceptibility has

been derived and the techniques appropriate for its evaluation described. We have shown

how the dynamic susceptibility is obtained from this by analytic continuation from Matsub-

ara frequencies in the complex plane to the real frequency axis [25]. This step provides a

natural interpretation of the spin dynamics in terms of overdamped oscillator models.

Although ultimately aiming to investigate the spin fluctuations in systems with complex

lattice structures such as Sr2RuO4 and MnSi which are of topical interest, here we have

compared and contrasted the nearly ferromagnetic transition metal Pd with the nearly anti-

ferromagnetic dilute Cr alloy, Cr95V5. In both cases we have been able to identify ‘slow’

spin fluctuations so that in due course the mode-mode coupling amongst them may be

incorporated into spin-fluctuational theories which describe the low temperature properties

of these materials [6,8]. For the case of Pd the spin susceptibility can be interpreted in

terms of two oscillator models. One describes spin fluctuations with fairly ‘fast’ relaxations

times but the other describes the spin fluctuations for all wave-vectors which have spin

relaxation times more than 40 times as slow as the time taken for an itinerant d-electron to

hop between lattice sites. Clearly there is a natural time scale separation between these spin

fluctuational modes and the electronic degrees of freedom. Consequently spin fluctuational

theory can be used with these modes without the need for any wave-vector cutoff. Work

is in progress to carry this out. On the other hand Cr95V5’s dynamic spin susceptibility is

interpreted in terms of a single oscillator. Here the spin relaxation times are only some 10

15

times slower than typical d-electron hopping times for modes with q’s in a limited region of

the Brillouin zone and therefore mode-mode coupling effects may not be so important. This

may explain why our calculations [10] of the static susceptibility of this Cr alloy (and Cr

and its other dilute alloys) receive an adequate description in terms of what is essentially

an ab-initio Stoner theory.

V. ACKNOWLEDGEMENTS.

We are grateful to S.Hayden and R.Doubble for useful discussions. JBS and BG also

acknowledge support from the Psi-k Network funded by the ESF Programme on ‘Elec-

tronic Structure Calculations for Elucidating the Complex Atomistic Behaviour of Solids

and Surfaces. JP acknowledges support from the Thailand Research Fund through con-

tract RTA/02/2542 and DDJ acknowledges support from the U.S. Department of Energy

grants DE-FG02-96ER45439 with the Fredrick Seitz Materials Research Laboratory and

DE-AC04-94AL85000 with Sandia.

16

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17

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19

FIGURES

FIG. 1. Imχ(q,q, w) of Pd at T =100K in units of µ2BeV −1 for wave-vectors q along the

0, 0, 1 direction where q is in units of π/a (a is the lattice spacing). The frequency axis, w, is

marked in meV. The full curve is for q =(0,0,0.02), then for curves with decreasing dash-length,

q = (0,0,0.1),(0,0,0.2) and (0,0,0.5).

FIG. 2. The variance of the spin fluctuations in Pd at T =100K, < m2(q) >, in µ2B for

wave-vectors q along the 0, 0, 1 direction with energy cutoffs of 500 meV (full line) and 50 meV

(dashed line).

FIG. 3. (a) The static susceptibility of Pd at T =100K partitioned into 2 parts χ1(q), χ2(q) of

equation (2.1) in units of µ2BeV −1 for wave-vectors q along the 0, 0, 1 direction. χ1(q) + χ2(q)

is shown by the full line, χ1(q) by the long-dashed line and χ2(q) by the short-dashed line. (b)

The time scales from eq. (2.1) (divided by h) in eV −1, Γ−11 (q) (full line), Γ−1

2 (q) (dashed line) for

Pd at T =100K. Note Γ−12 (q) has been scaled up by 100 for display purposes.

FIG. 4. The Fermi surface (FS) of Cr95V5 from Bloch spectral function calculations in the

qz = 0 plane. The dashed arrows mark the nesting vector qnest connecting the H-centered hole

octahedron to the slightly smaller Γ-centered electron surface. The FS is well defined with only a

small disorder broadening < 0.01π/a.

FIG. 5. The variance of the spin fluctuations in Cr95V5 at T =100K, < m2(q) >, in µ2B for

wave-vectors q along the 0, 0, 1 direction with energy cutoffs of 500 meV (full line) and 50 meV

(dashed line).

FIG. 6. The variance of the spin fluctuations in Cr95V5, < m2(q) >, at T =50K (full line),

300K (long-dashed) and 600K (short-dashed), in µ2B for wave-vectors q along the 0, 0, 1 direction

with a energy cutoff of 500 meV.

20

FIG. 7. (a) The static susceptibility of Cr95V5 at T =50K (full line), 300K (long-dashed) and

600K (short-dashed), in µ2BeV −1 along the 0, 0, 1 direction. (b) The time scale (divided by h) in

eV −1, Γ−1(q) for Cr95V5 at the same temperatures.

21

0 50 100 150 200w (meV)

0

1

2

3

4

5

6

7

Im X(q,w)

0 0.2 0.4 0.6 0.8 1q (0,0,1)

0

0.2

0.4

0.6

0.8

1

<m*m>

0 0.2 0.4 0.6 0.8 1q (0,0,1)

0

2.5

5

7.5

10

12.5

15

X(q)

0 0.2 0.4 0.6 0.8 1q (0,0,1)

0

100

200

300

400

500

600

relaxation time (/eV)

-0.4

-0.2

0

0.2

0.4

-0.5 0 0.5 1k

1 0 0( 2π/ a )

Γ H

N

N

k 01

0(2 π

/a)

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1q (0,0,1)

0

1

2

3

4

5

<m*m>

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1q (0,0,1)

3

3.5

4

4.5

5

5.5

6

<m*m>

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1q (0,0,1)

15

20

25

30

35

X(q)

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1q(0,0,1)

3

4

5

6

7

8

9

relaxation time (/eV)